ODE
\[ (a+x) y''(x)+(\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.268344 (sec), leaf count = 239
\[\left \{\left \{y(x)\to e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )} (a+x)^{a \text {b1}-\text {a1}+1} \left (c_1 U\left (\frac {-a \text {b1}^2+\text {a1} \text {b1}+a \sqrt {\text {b1}^2-4 \text {b2}} \text {b1}-2 \text {a2}+2 a \text {b2}-\text {a1} \sqrt {\text {b1}^2-4 \text {b2}}+2 \sqrt {\text {b1}^2-4 \text {b2}}}{2 \sqrt {\text {b1}^2-4 \text {b2}}},-\text {a1}+a \text {b1}+2,\sqrt {\text {b1}^2-4 \text {b2}} (a+x)\right )+c_2 L_{\frac {a \text {b1}^2-\text {a1} \text {b1}-a \sqrt {\text {b1}^2-4 \text {b2}} \text {b1}+2 \text {a2}-2 a \text {b2}+\text {a1} \sqrt {\text {b1}^2-4 \text {b2}}-2 \sqrt {\text {b1}^2-4 \text {b2}}}{2 \sqrt {\text {b1}^2-4 \text {b2}}}}^{-\text {a1}+a \text {b1}+1}\left (\sqrt {\text {b1}^2-4 \text {b2}} (a+x)\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.396 (sec), leaf count = 197
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {x \left (\mathit {b1} +\sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\right )}{2}} \left (a +x \right )^{\mathit {b1} a -\mathit {a1} +1} \KummerM \left (-\frac {\left (-\mathit {b1} a +\mathit {a1} -2\right ) \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}+a \,\mathit {b1}^{2}-2 a \mathit {b2} -\mathit {a1} \mathit {b1} +2 \mathit {a2}}{2 \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}}, \mathit {b1} a -\mathit {a1} +2, \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\, \left (a +x \right )\right )+\textit {\_C2} \,{\mathrm e}^{-\frac {x \left (\mathit {b1} +\sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\right )}{2}} \left (a +x \right )^{\mathit {b1} a -\mathit {a1} +1} \KummerU \left (-\frac {\left (-\mathit {b1} a +\mathit {a1} -2\right ) \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}+a \,\mathit {b1}^{2}-2 a \mathit {b2} -\mathit {a1} \mathit {b1} +2 \mathit {a2}}{2 \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}}, \mathit {b1} a -\mathit {a1} +2, \sqrt {\mathit {b1}^{2}-4 \mathit {b2}}\, \left (a +x \right )\right )\right ]\] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + (a1 + b1*x)*y'[x] + (a + x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((a + x)^(1 - a1 + a*b1)*(C[1]*HypergeometricU[(-2*a2 + a1*b1 - a*b1^2 + 2*Sqrt[b1^2 - 4*b2] - a1*Sqrt[b1^2 - 4*b2] + a*b1*Sqrt[b1^2 - 4*b2] + 2*a*b2)/(2*Sqrt[b1^2 - 4*b2]), 2 - a1 + a*b1, Sqrt[b1^2 - 4*b2]*(a + x)] + C[2]*LaguerreL[(2*a2 - a1*b1 + a*b1^2 - 2*Sqrt[b1^2 - 4*b2] + a1*Sqrt[b1^2 - 4*b2] - a*b1*Sqrt[b1^2 - 4*b2] - 2*a*b2)/(2*Sqrt[b1^2 - 4*b2]), 1 - a1 + a*b1, Sqrt[b1^2 - 4*b2]*(a + x)]))/E^(((b1 + Sqrt[b1^2 - 4*b2])*x)/2)}}
Maple raw input
dsolve((a+x)*diff(diff(y(x),x),x)+(b1*x+a1)*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-1/2*x*(b1+(b1^2-4*b2)^(1/2)))*(a+x)^(a*b1-a1+1)*KummerM(-1/2*((-a*b1+a1-2)*(b1^2-4*b2)^(1/2)+a*b1^2-2*a*b2-a1*b1+2*a2)/(b1^2-4*b2)^(1/2),a*b1-a1+2,(b1^2-4*b2)^(1/2)*(a+x))+_C2*exp(-1/2*x*(b1+(b1^2-4*b2)^(1/2)))*(a+x)^(a*b1-a1+1)*KummerU(-1/2*((-a*b1+a1-2)*(b1^2-4*b2)^(1/2)+a*b1^2-2*a*b2-a1*b1+2*a2)/(b1^2-4*b2)^(1/2),a*b1-a1+2,(b1^2-4*b2)^(1/2)*(a+x))]